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In high dimensional settings, sparse structures are crucial for efficiency,both in term of memory, computation and performance. It is customary toconsider $\ell_1$ penalty to enforce sparsity in such scenarios. Sparsityenforcing methods, the Lasso being a canonical example, are popular candidatesto address high dimension. For efficiency, they rely on tuning a parametertrading data fitting versus sparsity. For the Lasso theory to hold this tuningparameter should be proportional to the noise level, yet the latter is oftenunknown in practice. A possible remedy is to jointly optimize over theregression parameter as well as over the noise level. This has been consideredunder several names in the literature: Scaled-Lasso, Square-root Lasso,Concomitant Lasso estimation for instance, and could be of interest forconfidence sets or uncertainty quantification. In this work, after illustratingnumerical difficulties for the Smoothed Concomitant Lasso formulation, wepropose a modification we coined Smoothed Concomitant Lasso, aimed atincreasing numerical stability. We propose an efficient and accurate solverleading to a computational cost no more expansive than the one for the Lasso.We leverage on standard ingredients behind the success of fast Lasso solvers: acoordinate descent algorithm, combined with safe screening rules to achievespeed efficiency, by eliminating early irrelevant features.

Efficient Smoothed Concomitant Lasso Estimation for High Dimensional Regression